Theoretical model of a finite force at the moving contact line.

In early theoretical investigations of the moving contact line [H.K. Moffatt, Journal of Fluid Mechanics, 18, 1-18 (1964), C. Huh and L. E. Scriven, Journal of Colloid and Interface Science, 35 (1971)], an infinite force along the solid wall was reported based off the non-integrable stress along a single interface. Contrary to these results, this investigation demonstrates that while the stress is still singular, there is a finite point force at the moving contact line singularity if the forces applied along all three interfaces that make up the moving contact line are taken into consideration. Mathematically, this force is determined by summing all the forces that act over an infinitesimally small cylindrical control volume enclosing the moving contact line. With this finite force, we predict the microscopic dynamic contact angle based off a balance of forces at the moving contact line and subsequently combine our model with Cox's model for apparent dynamic contact angle [R.G. Cox, Journal of Fluid Mechanics, 168 169-194 (1986)]. This combined model is in good agreement with published dynamic contact angle measurements and eliminates the microscopic dynamic contact angle as an empirical fitting parameter. In support of our results, the proposed analysis also independently predicts the known forces of two other singular Stokes flows that contain stress singularities, namely the cusped fluid interface and Stokeslet.